a. \(\quad\) Mr. Odde Ball enjoys commodities \(X\) and \(Y\) according to the utility function $$U(X, Y)=\sqrt{X^{2}+Y^{2}}$$ Maximize Mr. Ball's utility if \(P_{x}=\$ 3, P_{Y}=\$ 4,\) and he has \(\$ 50\) to spend. Hint: It may be easier here to maximize \(U^{2}\) rather than \(U .\) Why won't this alter your results? b. Graph Mr. Ball's indifference curve and its point of tangency with his budget constraint. What does the graph say about Mr. Ball's behavior? Have you found a true maximum?

We will start by writing the budget constraint of Mr. Ball, given the prices of X and Y and his income. The budget constraint can be expressed as: $$ P_xX + P_yY = I $$ Substitute the prices and income into the equation: $$ 3X + 4Y = 50 $$

Now set up the utility maximization problem. Since it's easier to maximize U^2 instead of U (as hinted), and it won't alter the results, we'll work with U^2. Our objective is to maximize U^2 subject to the budget constraint: $$ \max_{X, Y} \quad U^2(X, Y) = X^2 + Y^2 \quad \text{subject to} \quad 3X + 4Y = 50 $$

To solve the maximization problem, we can use the method of Lagrange multipliers. The Lagrangian function is given by: $$ \mathcal{L}(X, Y, \lambda) = X^2 + Y^2 - \lambda(3X + 4Y - 50) $$ Now, we'll compute the first-order conditions by taking the partial derivatives with respect to X, Y, and λ: $$ \frac{\partial \mathcal{L}}{\partial X} = 2X - 3\lambda = 0 \\ \frac{\partial \mathcal{L}}{\partial Y} = 2Y - 4\lambda = 0 \\ \frac{\partial \mathcal{L}}{\partial \lambda} = 3X + 4Y - 50 = 0 $$ Solve the system of equations to find X, Y, and λ. We get: $$ X = \frac{3}{2}\lambda \\ Y = \lambda \\ 3\left(\frac{3}{2}\lambda\right) + 4\lambda = 50 \\ \frac{13}{2}\lambda = 50 \\ \lambda = \frac{100}{13} $$ Now, find the optimal values of X and Y: $$ X = \frac{3}{2}\lambda = \frac{3}{2} \times \frac{100}{13} = \frac{150}{13} \\ Y = \lambda = \frac{100}{13} $$

We have found that Mr. Ball maximizes his utility when consuming X = 150/13 and Y = 100/13. Now, draw the graph using the budget constraint and the equation of the indifference curve, U(X, Y) = k, where k is a constant: The budget constraint: $$ 3X + 4Y = 50 $$ The indifference curve: $$ U(X, Y) = \sqrt{X^2 + Y^2} = k \\ X^2 + Y^2 = k^2 $$ By plotting the budget constraint and the indifference curve passing through the optimal consumption point (150/13, 100/13), we can observe that they are tangent at this point, which confirms that we have found a true maximum. The graph also tells us that Mr. Ball is spending his entire budget on X and Y since the budget constraint is binding. Additionally, it shows that Mr. Ball's preferences exhibit diminishing marginal utility for both commodities, as the indifference curve is convex to the origin.